Harmonic Map Formulation of Colliding Electrovac Plane Waves

TL;DR

This paper reformulates the Einstein-Maxwell equations for colliding electrovac plane waves as harmonic maps, enabling a more straightforward approach to solving the initial value problem in such spacetimes.

Contribution

It presents three methods to express the Einstein-Maxwell equations as harmonic maps, facilitating the analysis of colliding plane wave solutions in general relativity.

Findings

Harmonic map formulations of Einstein-Maxwell equations are developed.

Initial data for colliding wave spacetimes are characterized.

Simplified harmonic map approach yields explicit metric and electromagnetic potentials.

Abstract

The formulation of the Einstein field equations admitting two Killing vectors in terms of harmonic mappings of Riemannian manifolds is a subject in which Charlie Misner has played a pioneering role. We shall consider the hyperbolic case of the Einstein-Maxwell equations admitting two hypersurface orthogonal Killing vectors which physically describes the interaction of two electrovac plane waves. Following Penrose's discussion of the Cauchy problem we shall present the initial data appropriate to this collision problem. We shall also present three different ways in which the Einstein-Maxwell equations for colliding plane wave spacetimes can be recognized as a harmonic map. The goal is to cast the Einstein-Maxwell equations into a form adopted to the initial data for colliding impulsive gravitational and electromagnetic shock waves in such a way that a simple harmonic map will directly yield the metric and the Maxwell potential 1-form of physical interest. For Charles W. Misner on his 60th birthday